PhysicsLettersA182 (1993)282—288 PHYSICS LETTERS ANorth-Holland

Nonlinearstationarywavesandsolitary wavesin fluid systemswith long-rangeforces

Fred C. Adams,MarcoFatuzzoandRichardWatkinsPhysicsDepartment,UniversityofMichigan,AnnArbor, MI 48109,USA

Received3 June1993;acceptedforpublication 17 September1993Communicatedby D.D. Holm

We studya classof nonlinearmodel equationswhichrepresentone-dimensionalfluid systemswith longrangeforces (suchasgravity) . We presentelementarytheoremswhich constraintheallowedtypesof stationarywaveandsolitarywavesolutionsforthisclassof systems.

1 . Introduction the gravitationalforcein onespatialdimensiondoesnot decreasein the asymptoticlimit x—~oo;on suf-

A wide variety of nonlinearwave structurescan ficiently largespatialscales,theinwardpull of gray-existin fluid systemswhereself-gravityis important, ity always dominatesand such systemsare thusbut suchmotionshavenotbeenwell studied.These doomedto collapse.wavesare most likely to occur in astrophysicalset- In this contribution,we studya generalclassoftings,althoughmanyotherphysicalsystemswith long modelequationswhichrepresentself-gravitatingfluidrangeforcescanmathematicallyresemblethisprob- systemsin onespatial dimension.In particular,welem (seebelow). In this Letter, we considera class showthatmanysystemscanbedescribedby theusualof nonlinearequationswhich model self-gravitating equationsof hydrodynamicscoupledwith a modi-fluid systemsin onespatialdimension.Ourgoal is fled form of the Poissonequationfor the gravita-to find general results which show how physical tional potential.Onemotivation for usinga modi-propertiesof such systemsdeterminethe allowed fled Poissonequationis to model two-dimensionaltypesof stationarywave solutions. gravitationaleffects(e.g., a finite extentof thewave

Fluid motionswhichsimultaneouslyincludeboth profile in thetransversedirection)while retainingaself-gravity andnonlineareffectscanexhibit inter- one-dimensionaltheory.Wepresentfourelementaryestingbehavior.In general,mostnonlinearwavesin theoremswhich constraintheallowedtypesof wavefluidstendto steepenandshock [ 1 ,2] . On theother behavioraccessibleto thesephysicalsystems.Thesehand,self-gravityleadstodispersion[3 ] whichtends results elucidatethe propertiesthat a systemmustto spreadout wavepackets.In principle,fluids with havein order to exhibit stationarywavesandsoli-self-gravity canreacha balancebetweennonlinear tarywaves.steepeningand gravitational dispersion.This bal-anceleadsto thepossibleexistenceofbothnonlinearstationarywaves and solitary wavesin the fluid. 2. General formulationHowever,previouswork in this area[4—6 1 hasshownthat the simplestpossiblesystem(i.e., eqs. ( 1 ) and In this section,we introduceaclassofmodelequa-(2 ) below coupledwith standardone-dimensional tions for the studyof nonlinearwavesin self-gravi-gravity) cannotexhibit solitary wavebehavior.The tatingfluid systems.Webeginwith the equationsofphysicalreasonfor this lackof solitarywavesis that motionforafluid. In onespatialdimension,thecon-

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Volume182, number2,3 PHYSICSLETTERSA 1 5 November1993

tinuity equationandtheforceequationtaketheformsv=u—vo, (6)

op a— + — (pu)= 0 , ( 1 ) which is simply the speedof thefluid relativeto theôt 8x speedv0 of thewave.Usingthesedefinitions,we in-

t9u t9u ~~ ~ ô~= 0 , (2) tegratethe continuity equation ( 1 ) to obtain the— +u_ +at ox p ox ~ relation

wherep isthe massdensity,u is thevelocity, ~uis the pvA , (7)gravitationalpotential, and p is the pressure.All wherethe constantof integrationA is the “Machquantitieshavebeenwritten in dimensionlessform. number”of the wave.We takethe pressureto havea generalbarotropic We canreducethe systemof equationsdescribedform aboveto a singlenonlineardifferentialequationfor

p=p(p) , (3) the densityp. If we differentiatethe force equation(2) withrespectto ~andusethegeneralizedPoisson

which includesmostequationsof stateof interest. equation(4) to eliminatethe potential,we obtainTo closethe systemof equations,we must include

3 c92p\

thePoissonequationfor thegravitationalpotential. ~ (p2 ~ A 2) +p~ 2 —p2 ~ +pIn thispaper,weconsidera classof systemswith thePoissonequationwritten in the generalizedform +p4q(p)=0 , (8)

~—,c= q(p) , (4) wherewehaveeliminatedthe velocity dependenceby usingthe solution (7) to thecontinuity equation

whereq(p) is afunctionof the densityp. Wedenote (andwheredifferentiation is now denotedby sub-the quantityq(p) asthechargedensity.Weusethis scripts).The equationof motion (8 ) is the funda-terminologybecause,in general,whateverappears mentalequationgoverningstationarywavesinchargeon theright handsideof a Poissonequationis often densitytheories.Fortunately,this highly nonlinearcalled“the chargedensity”. Notice that the charge equationcanbe integratedto obtaindensitydefinedhereis not the electric chargeden- (p2 ~ ...A2)f(p) ~ 3~(p), (9)sity. Notice also that the choiceq=pgives us back 2

the usualone-dimensionalPoissonequation.Equations(1 )—(4) completelydeterminethe sys- wherewehavedefinedf(p)tobeanintegralthatde-

ternoncetheequationof statep(p) andthe form of pendson the form of the chargedensityq(p), i.e.,thechargedensityq(p)arespecified.Wedenotethe-

q(p)(A2

ôp\oriesof this type as “chargedensitytheories”.For f(p) = J dp ~ —~ — i—) ~ (10)mostof this paper,wekeepthe forms forp(p) andq(p)arbitraryandpresentgeneralresults.In section Theexistenceor nonexistenceof solitarywavesand3, however,we show that many different physical stationarywavesfora particularphysicalsystemcansystemscanbe describedby chargedensitytheories beunderstoodthroughthestandardmethodsof phaseandwe presentspecific forms of q(p) for several planeanalysis[7,8 ] . In this methodwe reducetheexamples. systemto a singleequationof form (9 ). Theprop-

We focus this current discussionon stationary ertiesofthefunction ~F(p)determinethepropertieswaves,which correspondto travelingwavesof per- ofthewave solutionsp(c~)ina fairly simplemanner.manentform. For thesewaves,the fluid fields are Physicallymeaningfulsolutionsmusthave~ 0. Forfunctionsof the quantity the systemsconsideredhere,the field p is a mass

densityand must alwaysbe positive. Thus, physi-c~=x—v

0t, (5)cally relevantsolutionsexist when~F(p) is positive

wherev0 is the (dimensionless)speedof the wave. overa rangeofpositivedensities.Wavesolutionsex-Next, we introducea new velocity variable ist when~3~(p)is positivebetweentwo zeroesofthe

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function .~, where the zeroescorrespondto maxi- In particular,we write the Poissonequationin themumandminimumdensitiesofthewaveprofile. The generalizedformnatureof the zeroesof ~Fdeterminesthe natureofthe wavelikesolutions.Forexample,if 3~is positive 2 = m2yJ+p. (14)betweentwo simplezeroes,then (ordinary) nonlin- ~ear wavesresult.However, if ~ is positivebetween This theory producesan exponentialfalloff in thea simplezeroanda doublezero (whereboth~ and gravitationalforcebetweenpointmasses;theparam-ô3~~/ôpvanish),thensolitarywave solutionscanbe eterm determinesthe effective rangeof the force.found [ 7—10] . Theright handsideof eq. (14) definesan effective

chargedensityq, althoughit is notwritten explicitlyas a function of densityonly. However,we canin-

3. Examplesof chargedensitytheories tegratethe stationaryversionof the force equation(2) to obtain

As weshowin thissection,manyphysicalsystemsyi+h(p)+~v2=E, (15)can be describedby charge density theories. The

simplestexampleof a nontrivial chargedensity is whereE is theconstantof integration.Thequantitygivenby h(p) is the enthalpyandis definedby

p

q(p)=p—p0, (11) h(p)=$

4~. (16)wherePo is the dimensionlessbackgrounddensityofthe fluid. Wecanobtainthis chargedensityby sub- The speedv canbe eliminatedby usingthe solutiontractingoutthe contributionto the gravitationalpo- to the continuity equationfor stationarywaves,i.e.,tential due to the backgroundfluid; this charge

V= A/p. Usingtheseresultsin eq. ( 1 4 ), wecanreaddensitytheorythusreproducesthe original approx- off the form of the chargedensity,i.e.,imation of Jeans [ 3,1 1 ] . Anotherway to obtain atheorywith the abovemathematicalform is to con-sidera physical systemwhich rotatesat a uniform q(p)=~ + 2 (E_ h(p) — ~). ( 17)rateQ [2 ] . In the directionperpendicularto the ro-tation axis, Gauss’slaw for a uniform density cyl- Themotivationfor introducingthe extratermin theinderwith densityPo implies Poissonequation(14) is to allow the gravitational

forceto decreasewith increasingdistance.As weshowaiQ2= 2icGp

0w , ( 1 2 ) in ref. [ 12 ] , anotherway to obtainthis generalbe-

whereW is theradialcoordinateandwhereall quan- havior is to reducea two-dimensionaltheoryto antitieshavetheirusualdimensions.Thus,theuniform effectiveone-dimensionaltheoryusingratherseveredensitystatewill be in mechanicalbalanceprovided approximations.In particular,we assumethat thethat velocity in the transversedirection is small com-

paredto that in thedirectionof propagationandwePo Q

2/21tG. ( 1 3 ) considera simplified treatmentof the wave profilein the transversedirection.We canthus obtain yetFor wavespropagatingalongthe axis of rotation inanotherchargedensitytheorywith

this system,we obtain equationsof motion with a“chargedensity” of form ( 1 1 ) withPo givenby eq. ~9p 18)(13). q(p)=p—2~2

Wecanalsoderivea chargedensityfor theoriesinwhichgravity is modeledby a Yukawapotential.The where~ is a parameterwhich determinesthe steep-motivationfor this approximationis to incorporate nessof the wave profile in the transversedirection.a decreasinggravitational field strengthwhile re- The chargedensity ( 1 8 ) is qualitatively similar totaminga one-dimensionaltreatmentof theproblem. that obtained from a Yukawa potential (see eq.

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( 17) ) andtheparameters1u andm playa similarrole self-consistently.The net result of theseapproxi-(seeref. [ 1 2 1 for further discussion) . mationsis to makethemagneticforceontheneutral

Thestandardproblemofvolumedensitywavesin componentinto a long-rangeforcethatbalancesthea plasmacanalso be written in terms of a charge long-rangeforce of gravity. This treatmentis un-densitytheory.Theequationsof motion governing physicalin thesensethatthesemagneticforces,whichthedynamicsoftheionsarethesameaseqs.( 1 ) and arisefromthe frictional forceexertedon theneutral(2) with the electricpotential0 replacingthe graY- componentsby the ions, are intrinsically local. Onitational potential yi. ThePoissonequationfor the theotherhand,this approximationallows forsome-electric potential (in dimensionlessform) canbe thingto cancelthe long-rangeforceofgravity,whichwritten doesnot fall off with distancebecauseof the one-

2 dimensional treatmentof the problem (see refs.

~ = exp(çb) —p , ( 1 9 ) [4,1 2 ] for further discussion).

wherep is now theion density[ 1 3,7 ] . Proceedingaswe did for the Yukawapotential,we canintegrate 4, General theorems and resultsthe stationaryversionof the force equationto findthe potential andthenuse the solution to the con- In this sectionwe presentgeneralanalyticresultstinuity equationto eliminatethe velocity. We thus thatare applicablefor arbitraryformsof the chargeobtain a chargedensitytheorywith densityq(p). In particular,we presentfour elemen-

2 2 tary theoremswhich greatly constrainthe allowedq(p)=exp[E—h(p)—A /2p ]—p (20) . .typesof wavebehaviorfor thisclassof theones.De-

whereE istheconstantofintegrationandh(p) isthe tailed proofsof theseresultsandfurther discussionenthalpy(thepressureis oftentakento benegligible will be givenin ref. [ 12].in this problemandhenceh= 0). To begin,we presenta theoremwhich showsthat

Anotherway to obtain a chargedensitytheory is for any theorywith a chargedensityq(p), a strongtosimulatetheeffectsofanembeddedmagneticfield. constraintmustbe met inorderforstationarywavesIn ref. [4], wederiveda model equationon thisba- to exist. This constraintarisesfrom the fact that asis. We assumedthat the magneticfield points in a stationarywave musthavelocal extremawherethedirectionperpendicularto that of the wave motion pressureandvelocity gradientsgo to zero. There-andthat the neutralcomponentof the fluid is cou- fore, in orderfor thewave to remainstationary,thepledto the magneticfield throughits frictional in- forceofgravitymustalsovanishat thesepoints.Thisteractionwith the ionized fluid component(which behaviorcanonly occurif the integralof the chargeis well coupledto the field). We also ignored any densityq(p) betweentwo extremais zero.This ar-chemicaleffectssothat the ionizedcomponentofthe gumentcanbeput in the formof a theoremforsta-fluid obeys a continuity equation. The resulting tionary waves.model equationcanbe derivedfrom a theorywiththe chargedensitywritten in the form Theorem1 . If a stationarywavesolutionp(c~)cx-

I \ istsfor the one-dimensionaltheory,thentheintegral

q(p)=p+r’( ~— ~i.~I (21 ) ofthechargedensityoveronewavelengthmustvan-

\Jo PFJ ish, i.e.,whereI’ representsthe couplingstrengthbetweentheneutraland ionizedcomponentsandwherePF de- Q~ J d~q[p(c~)] = 0,terminestheion density (assumedtobe constantinref. [4] ). Wenote that the model correspondingtothe chargedensity (21 ) is idealized in two ways. wherethe wavelengthof a solitarywave is takentoFirst, a dissipativetermhasbeendroppedto obtain be infinite. This claim is limited to the caseof non-this form. Second,the ion density is not calculated singularsolutions;for solitonswe also requirethat

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the densitydoesnot vanishat spatial infinity. We first find a necessaryconditionon the charge

Theorem1 greatlylimits theallowedtypesofwave densityfor solitary waves.When suchwavesexist,behaviorin systemswith self-gravity. For the sim- the first integral .~ of the equationof motion mustplestcaseof q(p)=p, theorem1 impliesthatno sta- bepositivebetweena doublezero anda ordinaryzerotionary wavesor solitary wavescanexist becausep ~seeeqs. (9 ) , ( 10 ) ) . We restrictour discussiontois a massdensityandis positivedefinite (the ab- waveswhich avoidthe singularityat the sonicpointsenceof solitarywavesin thiscontextwasfoundpre- (whichoccurswhenthetermin largeparenthesesinviously in bothrefs. [4 ] and [ 5 ] ). Anotherimpor- eq. (9 ) vanishes).Wealso restrictour discussiontotantconsequenceofthis “no-chargetheorem”is that waveswith finite velocity; this secondrestrictionthe chargedensityq(p) mustvanishin the asymp- eliminatesthepointp= 0 asa candidatefor thedou-totic limit ~—~oofor a solitary wave or soliton. For ble zero (noticethat for~o—÷O~weget v=A/p—~oobysolitarywaves,thewavelengthis infiniteandthemass the continuity condition(7 ) ) . Therequireddoubledensityp(t~)approachesa constantas ~—~oo. zeroof ~ mustthusoccurat a doublezero off(p).

Next,weconsidertherelationshipbetweentheex- Furthermore,f(p ) mustbe a localminimum at theistenceof solitary wavesand the absenceof gravi- doublezero.In orderfor thesecond(ordinary) zerotational instability. This relationshiparisesbecause of ~Ftoexist,thefunctionf(p) mustalsohavealocala solitary wave is, in somesense,an infinite wave- maximumat someotherdensity.By definition (eq.length perturbationon a uniform medium. On the ~10) )‘ thederivativeoff(p) is proportionalto q(p).otherhand,the presenceof gravitationalinstability Sincewe mustavoidthesingularityat thesonicpoint,implies that all perturbationswith sufficiently long both the requiredminimum andthe maximumofwavelengthswill collapse[ 3 ] . Thus,the existenceof .flP) mustoccurwherethe chargedensityq(p) van-a stationaryperturbationwith an infinite wave- ishes.We denotethe locationof the minimumas1c”length (a solitary wave) is inconsistentwith the andthemaximumasPM. In orderfor thefirst criticalpresenceof gravitational instability. This relation- point to be a minimum andfor the secondto be ashipcanbe statedmorepreciselyas follows. maximum,wemusthavethecorrectsignsfor dq/dp

at the critical points (where the signsdependonTheorem2. Supposea physical systemobeysa whetherwe considersubsonicor supersonicwaves).

generalizedmodel equationof motion of form ( 8 ) Thisargumentcanbe summarizedas the followingandthis model equationhassolitarywavesolutions. necessaryconditionon the chargedensityq(p) forThenthe systemcanhavea uniform densitystate the existenceof solitarywaves.thatisgravitationallystableforarbitrarilylargelengthscales. Theorem3. In orderfor solitary wavesolutionsto

exist for the classof theoriesconsideredin this pa-Wenotethattheconverseoftheorem2 is nottrue. per, thechargedensityq(p)musthave(atleast)two

Physicalsystemsof this type canbe gravitationally zeroesp~andPM. For subsonicwaves,q(p)mustbestable to perturbationsof all wavelengthsandstill negative betweenthe two zeroes; for supersonicnot havesolitary wave solutions.Thus,the require- waves, q(p) mustbe positivebetweenthe zeroes.mentof gravitationally stableconfigurationsrepre- (Thisresultappliesonly to nonsingularsolutionsforsentsa necessarycondition (and not a sufficient which the sonicpointPcdoesnot lie betweenPi andcondition) for the existenceof solitary waves. PM

The questionof whetheror not solitarywave andstationarywave solutionsexistisfundamentalto the Theorem3 is potentiallyverypowerfulasa testtostudyof nonlineardynamics.For the classof charge seeif solitary wave solutionsexist. For example,density theoriesconsideredin this paper,the ques- manypossiblechargedensityfunctionsdo nothavetion becomes:What propertiesof the chargedensity two zeroesandthus solitary wavebehaviorcanbeq(p) are requiredfor the existenceof solitary waves easily ruledout. We note, however,that theorem3andwhatpropertiesarerequiredfor theexistenceof is notsufficienttoguaranteetheexistenceofsolitarystationarywaves? waves.We mustplacean additional constrainton

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the chargedensityto makesurethatthe second(or- 5. Summaryand discussiondinary) zero of the functionf(p) exists. Giventhatq(p) hasat leasttwo zeroes,the functionf(p) has In thisLetterwe havedevelopedmethodsto studythreepossibleforms.Thefirst possibilityis thatf(p) wavemotionsin self-gravitatingfluids inonespatialhasa secondordinaryzero andhenceordinarysol- dimension. In particular,we have introducedtheitary wavesexist. The secondpossibilityis that the conceptof”chargedensity”for thestudyofsuchsys-function f has anotherminimum at some density tems.In this formalism,themassdensityis replaced

p3>PM andhenceturnsup at largedensities;in this by a “chargedensity”q(p) on theright handsideofcasewe canalwayschoosethe constantof integra- the Poissonequationfor the gravitationalpotentialtion in eq. ( 10) to makep3 a double zerooff We (the continuityequationandthe forceequationre-thusobtaina depressionsolution or voidsolution.For main asusual).We haveshownthatmanyphysicalthe remainingpossibility, f(p) asymptoticallyap- systemscanbe modeledusingchargedensitytheo-proachesa constantandthesystemfails to havesol- ries. In addition, we haveshownthat the standarditary wave solutions.This issueof the existenceof exampleof volume densitywavesin a plasmacanthe secondzerooff(p) is discussedmorefully in ref. alsobewritten asa chargedensitytheory(wherethe[ 12 ] . electric potential replacesthe gravitationalpoten-

Next,weconsiderthe simplercaseofordinarysta- tial). We canthusstudya wide rangeofphysicalsys-tionarywaves.Therequiredconditionsonthecharge temsandphysicaleffectsusingthis method,whichdensityq(p) for the existenceof thesewavescanbe resultsin a simplesemi-analytictheory.statedasfollows. Wehavepresenteda “no-chargetheorem”(theo-

rem 1 ) which showsthat no solitary wavesor sta-Theorem4. In orderforstationarynonlinearwave tionarywavescanexistin one-dimensionalself-gray-

solutionsto existfor theclassof theoriesconsidered itating fluids unlessthe totalchargevanishes(wherein thispaper,the chargedensityq(p)musthave(at the total chargeis the integralof the chargedensityleast)onezeroPM. Forsubsonicwaves,dq/dp> 0 at overonewavelength).This requirementof vanish-

PM andthiszero mustoccurat a densitylargerthan ing totalchargegreatlyconstrainsthetypesof modelthe sonicdensityPc (the densityof the singularity). equationsthatallow for stationarywave behavior.Similarly, for supersonicwaves,thesystemmusthave Wehaveshownthat in orderfora physicalsystemdq/dp< 0 at PM<Pc• to exhibit solitary wave behavior,the systemmust

alsobe capableof a configurationwhich is gravita-The conditionsoutlined in theorem4 are both tionally stable to perturbationsof arbitrarily large

necessaryandsufficientfor the existenceof station- wavelengths(seetheorem2). We havethusdiscov-ary waves.This result is straightforwardto under- ered a fundamentalrelationshipbetweengravita-stand.In orderfor suchwavesto exist, the function tional stability andthe existenceof solitary waves.f(P) musthavea maximumandtwo zeroesp’ <P2 WehavefoundnecessaryandsufficientconditionssuchthatPc<Pi for subsonicwavesandPc>P2for on the form of the chargedensityq(p) for the cx-supersonicwaves.The zero of q(p) is requiredto istenceof solitary waves and ordinary stationaryprovide a critical point for f(p) . The sign require- waves(seetheorems3 and4 ) . Theseconditionsal-ment on dq/dpensuresthat the critical point PM is low usto determine(or at leastconstrain)thepos-a maximum. The third condition, that the critical sible typesof wave behaviorfor any chargedensitypointPM be largerthan the sonicpoint for subsonic theorywithouthavingto solvetheequqtionsofmotion.wavesand smallerthan the sonic point for super- In summary,we haveintroduceda new classofsonic waves,is requiredsothat the singularitydoes “chargedensitytheories”for the studyof wavemo-not lie in the rangeof densitiesof the wave profile. tions in self-gravitatingfluids andrelatedsystems.

Wehavearguedthat many physicalsystemscanbemodeledwith chargedensity theories.Finally, wehaveprovengeneralresults (theorems1—4 ) whichprovidepowerfulconstraintson theallowedtypesof

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wavebehaviorin thesesystems.In a forthcomingpa- Referencesper [ 1 2 1 , wepresentproofsofthe theoremsandusetheresultsto studyseveralexamplesof chargeden-sity theories. ~1 ] G.B. Whitham, Linearand nonlinearwaves(Wiley, New

York, 1974).Oneprimary motivationof this work wasto un-

[2] F.H. Shu, Gasdynamics(University ScienceBooks, Millderstandwavemotionsinself-gravitatingastrophys- Valley, 1992).

ical systems,suchasstar forming regionsin molec- ~3 ] J.H.Jeans,Astronomyandcosmogony(CambridgeUniv.

ularclouds [ 12,14].However,otherphysicalsystems Press,Cambridge,1928).

can be modeledwith chargedensity theories(see, [4] F.C.AdamsandM. Fatuzzo,Astrophys.J.403 (1993) 142.

e.g.,eq.(20) andtheresultsofthispapershouldthus [5] E.P.T.Liang, Astrophys.J. 230 ( 1979)325.[6 } G. Götz,Class.QuantumGray. 5 ( 1988) 743.

beapplicableto manyotherproblems.In additionto [7] E. Infeld andG. Rowlands,Nonlinearwaves,solitons,and

variousdirectapplicationsof theseresults [ 1 2 ] , fur- chaos(CambridgeUniv. Press,Cambridge,1990).

ther work shouldbe doneon generalchargedensity [8] PG. DrazinandR. S. Johnson,Solitons:an introduction

theories.In particular,therelatedissuesof wavesta- (CambridgeUniv. Press,Cambridge,1989).

bility [ 1 5 1 and wave excitation [ 1 6 ] should be [9] S. Coleman,Aspectsofsymmetry(CambridgeUniv. Press,Cambridge,1985).

addressed.[ 1 0 ] R. Rajaraman,Solitonsandinstantons:anintroductionto

solitonsand instantonsin quantumfield theory (North-Holland,Amsterdam,1987).

Acknowledgement [ 1 1 ] J.Binney andS. Tremaine,Galacticdynamics(PrincetonUniv. Press,Princeton,1987).

Wewould like to thankIraRothsteinandMichael [12] F.C. Adams,M. Fatuzzo and R. Watkins, submitted to

Weinsteinfor usefuldiscussions.Thiswork wassup- Astrophys.J. ( 1993).

ported by NASA Grant no. NAGW-2802 and by ~13] R. C. Davidson, Methods in nonlinearplasma theory(AcademicPress,NewYork, 1972).

funds from the PhysicsDepartmentat the Univer- ~14] F.H. Shu, F.C.AdamsandS. Lizano, Annu. Rev.Astron.

sity of Michigan. MF is supportedby a Compton Astrophys.25 ( 1987)23.

GRO Fellowship from NASA. I 1 5 ] R.L. Pego and MI. Weinstein,Phys.Lett. A 1 62 ( 1992)263.

[l6}J. AronsandC. Max, Astrophys.J. Lett. 196 (1975)L77.

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